In this paper, we give a direct proof of Rockafellar's result that the subdifferential of a proper convex lower-semicontinuous function on a Banach space is maximal monotone. Our proof is simpler than those that have appeared to date. In fact, we show that Rockafellar's result can be embedded in a more general situation in which we can quantify the degree of failure of monotonicity in terms of a quotient like the one that appears in the definition of Fréchet differentiability. Our analysis depends on the concepts of the least slope of a convex function, which is related to the steepest descent of optimization theory.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Rockafellar, R. T.,On the Maximal Monotonicity of Subdifferential Mappings, Pacific Journal of Mathematics, Vol. 33, pp. 209–216, 1970.
Taylor, P. D.,Subgradients of a Convex Function Obtained from a Directional Derivative, Pacific Journal of Mathematics, Vol. 44, pp. 739–747, 1973.
Borwein, J. M.,A Note on ε-Subgradients and Maximal Monotonicity, Pacific Journal of Mathematics, Vol. 103, pp. 307–314, 1982.
Phelps, R. R.,Convex Functions, Monotone Operators, and Differentiability, Springer-Verlag, Berlin, Germany, 1989.
Brøndsted, A., andRockafellar, R. T.,On the Subdifferentiability of Convex Functions, Proceedings of the American Mathematical Society, Vol. 16, pp. 605–611, 1965.
König, H.,Some Basic Theorems in Convex Analysis, Optimization and Operations Research, Edited by B. Korte, North-Holland, Amsterdam, Holland, pp. 107–144, 1982.
Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley, New York, New York, 1983.
Burke, J. V., andMoré, J. J.,On the Identification of Active Constraints, SIAM Journal on Numerical Analysis, Vol. 25, pp. 1197–1121, 1988.
Dem'yanov, V. F., andMalozemov, V. N.,Introduction to Minimax, John Wiley, New York, New York, 1974.
Gale, D.,A Geometric Duality Theorem with Economic Applications, Review of Economic Studies, Vol. 34, pp. 19–24, 1967.
The author would like to express his thanks to R. R. Phelps for reading a preliminary version of this paper and making some very valuable suggestions.
Communicated by O. L. Mangasarian
About this article
Cite this article
Simons, S. The least slope of a convex function and the maximal monotonicity of its subdifferential. J Optim Theory Appl 71, 127–136 (1991). https://doi.org/10.1007/BF00940043
- Subdifferential of a convex function
- maximal monotone operator
- separation theorem
- sandwich theorem